What is the quadratic equation and what does it do?
In: 287
The simplest “non linear” function is a quadratic function, any function that is of the form y = ax^2 + bx + c with a b c being anything you choose. This makes kind of a U shape, we also call these parabolas.
It’s sometimes important to figure out for which x values this function hits y = 0, which it often does twice, and generally important to find where it hits any y value. We can use the quadratic formula to figure out where this happens based on a, b, and c.
One common example is throwing things. Throwing anything follows a ballistic (quadratic) path, if you know how strong you threw something, you can use the quadratic formula to figure out where and when it will hit the ground.
The quadratic equation is a mathematical formula that helps us find the values of x in a quadratic function
It’s like a secret code to unlock the solution to a quadratic problem.
An equation is something with an = in it. One thing (or set of things) is the same as another thing.
Usually equations will involve unknowns (things we don’t know but want to find out) and things we do know, and the goal is to use them (and the various mathematical rules we have) to figure out what the unknowns are or could be.
The simplest equations we have are linear equations, they look like:
> ax + b = 0
We have a constant term (b) and an “x” term multiplied by some constant (a) scale factor. We can always solve this by rearranging to get a single answer for “x”:
> x = -b/a
The next simplest kind of equations we get are where we also throw in an x^2 term:
> ax^2 + bx + c = 0
we have our constant, our linear term, and now a quadratic term in that “x^(2).”
Quadratic equations – being one of the most basic forms of equation we can get – crop up all over the place. Particularly with things that change. Anything that changes at a constant rate gives us a linear equation. Anything that changes at a rate that changes at a constant rate gives us a quadratic.
Because they turn up so often, and are relatively simple, it is worth it for us to just memorise the solutions, or solve them in their most general form.
And the neat thing about quadratic equations is that there is a *general solution* to them. Given any constants a, b, and c (with a not being 0) we can find solutions:
> x = -b/2a + sqrt(b^2 – 4ac)/2a
and
> x = -b/2a – sqrt(b^2 – 4ac)/2a
No matter the values of a, b and c, these values will solve our equation.
We can also look at quadratic functions:
> f(x) = ax^2 + bx + c
As functions these things will take different values depending on what we put in as “x.” Here “x” isn’t an *unknown* but a *variable*; it varies, taking a bunch of different values. And for each possible x we put into our function we will get out a specific value for *f*.
Because quadratics turn up so often these are also worth studying in the most general form; we can look at their behaviour for different values of a, b and c. We can look at the patterns they have, the kinds of solutions we get for *f(x) = k* and so on.
Quadratics are neat equations; they provide a certain non-trivial level of complexity, while also being completely solvable in general terms. Which also makes them great for teaching students key mathematical ideas.
Come on folks. Break this down like I’m 5. I failed algebra and want to understand.
Imagine you have a big slide on the playground, and you want to know when you’ll reach the bottom. The quadratic equation helps you figure that out when you know how fast you’re sliding (that’s one number) and how tall the slide is (that’s another number). It’s like a magical formula that tells you exactly when you’ll reach the ground while sliding down the slide! That’s about how ELI5 I can get…
[Let’s say you want to know what the area of the red circle is.](https://i.imgur.com/LaztZsW.jpg)
You need to use a quadratic equation. I only solved this one because none of the Facebook users would tell me what the answer was.
A lot of things in life can be solved by mathematical formulas.
For instance, let’s say you’re traveling at 60 miles per hour and want to know how long will it take you to travel 180 miles.
Simple: 180 miles / 60 mph = 3 hours
Or more generally: X miles / Y mph = Z hours
There are a *lot* of formulas like this that humans have discovered and use to calculate and predict things (e.g. where will this rocket land if we launch it at this angle? where will the moon be at this time? etc). Many of these formulas happen to follow this pattern:
ax^2 + bx + c = 0
This pattern comes up frequently enough that we decided to just memorize the solution for finding x and give that formula a name we could refer to it as: the quadratic formula.
Quadratic equations pop up a lot when you have a situation where something is moving and you want to find out things like how high/low something is at a point in time or what time it hits the ground/reaches a particular destination, or at what speed it has at a point in time.
For example, if you throw a ball into the air, how high does it go? What time does it hit the ground? Quadratic equations let you find out the answer to these questions.
Let’s try a simpler explanation and start at a little more basic point.
When two things are related, you can express their relation using an equation. The type of equation tells us how they are related. For example, how much money you spend when you buy 1 item vs. 10 items is linear. Just multiply both the cost and the item count by 10 and you get the answer. How far can you travel at a constant speed in one hour vs. 2 hours? All of these are called linear relationships.
Some relationships are a little more complex. For example, what if the speed itself changes? If you apply constant acceleration continuously for an hour, the speed increases, because speed is linearly related to acceleration. The distance is now linearly related to something that is itself linearly increasing. This makes the distance related to the square of time. This is now called a quadratic equation. That is all that defines a quadratic equation – the unknown quantity changes based on the square of the known quantity. You can have a linear component and a constant component added, but as long as there is a square component and nothing higher (third power, etc.) it will be a quadratic equation.
The quadratic equation is useful because it describes how these quantities change and help predict them. There are a lot of these relationships found in nature and mathematics. A lot of motion is based on solving quadratic equations, so it helps us plan and predict everything from missiles and space launches to calculating square footage in a home.
It provides a programmatic way of solving equations of the form Ax^2 + Bx + C which are very important for differential equations in oscillating or damped systems which are critical to understanding engineering. Read about Bode plots.
Latest Answers